Circles and Angle Properties
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Circles and Angle Properties — Free MYP4 Mathematics (Standard) Practice Questions

1QuestionAngles at the Centre and in the CircleConcept Practice
4 marks~6 minCriterion A
An architect is designing a circular skylight. Points AA, BB, and CC lie on the circumference of a circle with centre OO. The architect measures ∠BAC=34°\angle BAC = 34°. The central angle ∠BOC\angle BOC is modelled by the expression (2x+10)°(2x + 10)°.
a
Deduce the measure of ∠BOC\angle BOC. [1]
b
Find the value of xx. [1]
c
The architect states: "Because the central angle exceeds 60°60°, arc BCBC spans more than one-sixth of the full circle." Justify whether this statement is correct. [2]
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Solutions

2QuestionCyclic Quadrilaterals and ChordsAssessment Practice
4 marks~6 minCriterion D
A horticulturalist designs a circular greenhouse whose floor plan is modelled as a cyclic quadrilateral ABCDABCD with side lengths aa, bb, cc, dd and semi-perimeter s=a+b+c+d2s = \dfrac{a+b+c+d}{2}. The area is estimated using Brahmagupta's formula:

Area=(s−a)(s−b)(s−c)(s−d)\text{Area} = \sqrt{(s-a)(s-b)(s-c)(s-d)}
a
Explain why Brahmagupta's formula requires ABCDABCD to be a cyclic quadrilateral, referring to a key angle property. [1]
b
Discuss one strength and one limitation of using this formula to estimate the usable growing area of a real greenhouse. [2]
c
The horticulturalist measures the constructed greenhouse and finds the actual dimensions differ from the design. Advise the horticulturalist whether Brahmagupta's formula remains a reliable basis for calculating growing area, and recommend one specific adjustment to the design process. [1]
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Solutions

3QuestionAngles in the Same Segment and Alternate Segment TheoremAssessment Practice
4 marks~6 minCriterion C
A circular road junction is modelled by a circle with four points AA, BB, CC, and DD on its circumference, forming cyclic quadrilateral ABCDABCD. A traffic engineer records the following angles:

∠BAC=(3x+10)∘,∠BDC=(5x−30)∘,∠BCD=(4y)∘\angle BAC = (3x + 10)^\circ, \quad \angle BDC = (5x - 30)^\circ, \quad \angle BCD = (4y)^\circ
a
Explain why ∠BAC=∠BDC\angle BAC = \angle BDC. [1]
b
Deduce two equations in xx and yy using the properties of cyclic quadrilaterals. [1]
c
Solve your equations to find xx and yy, showing all working. [1]
d
Calculate ∠BAD\angle BAD and justify whether the engineer's claim that the layout forms a valid cyclic quadrilateral is correct. [1]
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Solutions

4QuestionCyclic Quadrilaterals and ChordsAssessment Practice
6 marks~9 minCriterion B
An architect is designing a circular plaza. Four anchor points AA, BB, CC, DD lie on the circle, forming cyclic quadrilateral ABCDABCD. A diagonal support beam runs along chord ACAC.

The following measurements are recorded:

ABAB (cm): 5, 8, 7, 6, 5

BCBC (cm): 5, 8, 7, 8, 12

∠ADC\angle ADC (degrees): 60, 120, 90, 60, 90
a
Deduce a general formula for the length of chord ACAC in terms of ABAB, BCBC, and ∠ADC\angle ADC. [3]
b
Calculate the length of chord ACAC for each set of measurements. [2]
c
The support beam ACAC can only be installed if its length is strictly less than AB+BCAB + BC. Justify, for each case, whether the beam can be installed, and explain what geometric property guarantees this will always be the case. [1]
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Solutions

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